60.10.14 problem 1926

Internal problem ID [11925]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1926
Date solved : Tuesday, January 28, 2025 at 06:24:08 PM
CAS classification : system_of_ODEs

\begin{align*} x \left (t \right )&=t \left (\frac {d}{d t}x \left (t \right )\right )+f \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right )\\ y \left (t \right )&=t \left (\frac {d}{d t}y \left (t \right )\right )+g \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.519 (sec). Leaf size: 95

dsolve([x(t)=t*diff(x(t),t)+f(diff(x(t),t),diff(y(t),t)),y(t)=t*diff(y(t),t)+g(diff(x(t),t),diff(y(t),t))],singsol=all)
 

\begin{align*} \{\int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_{1} &= \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right ) t +f \left (\operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right ), \frac {d}{d t}y \left (t \right )\right )\} \\ \{x \left (t \right ) &= \int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_{1}\} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 28

DSolve[{x[t]==t*D[x[t],t]+f[D[x[t],t],D[y[t],t]],y[t]==t*D[y[t],t]+g[D[x[t],t],D[y[t],t]]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to f(c_1,c_2)+c_1 t \\ y(t)\to g(c_1,c_2)+c_2 t \\ \end{align*}